Finite Group Databasebeta

← #101356

Subgroups — #101356

Order 512.

392 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27685C211yesmaximal, characteristic
2#22325C211yesmaximal, characteristic
3#8717C211yesmaximal, characteristic
4#8710C211yesmaximal, characteristic
5#8576C211yesmaximal, characteristic
6#7044C211yesmaximal, characteristic
7#7024C211yesmaximal, characteristic
8C4×C22⋊Q8C2211yescharacteristic
9#2285C2211yescharacteristic
10#2100C411yescharacteristic, complemented
11#2099C411yescharacteristic, complemented
12#2096C411yescharacteristic, complemented
13#2095C411yescharacteristic, complemented
14#1985—21no—
15#1949—21no—
16#1924—21no—
17#1924—21no—
18#1917—21no—
19#1916—21no—
20#1873—21no—
21#1854—21no—
22#1817C2211yescharacteristic
23#1805—21no—
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26#1771—21no—
27C4×C22⋊C8C2211yescharacteristic
28C2×#328—21no—
29#1434C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1279—21no—
34#1279—21no—
35C2×C22⋊Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C2×#117—21no—
38C22×C42C2311yescharacteristic
39C2×#107—21no—
40C2×#106—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8D411yescharacteristic
51C2×C22⋊C8D411yescharacteristic
52C2×#97—21no—
53C2×#97—21no—
54C2×C4×C8D411yescharacteristic
55C2×C4×C8D411yescharacteristic
56#391C2×C411yescharacteristic
57#389C2×C411yescharacteristic
58#387—21no—
59#385—21no—
60#384C2×C411yescharacteristic
61#384C2×C411yescharacteristic
62#382—21no—
63#380—21no—
64#380—21no—
65#380D411yescharacteristic
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4D411yescharacteristic
70C4×C22⋊C4—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#328—42no—
89C22×Q8C22⋊C411yescharacteristic
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—41no—
101C22⋊Q8—21no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C4×Q8—41no—
106C4×Q8—41no—
107#117—81no—
108#117—81no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—41no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133#107—81no—
134#106—81no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—42no—
140C4⋊C8—81no—
141#103—81no—
142#103—81no—
143C22⋊C8—42no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#97—42no—
151#97—81no—
152C4×C8—41no—
153C4×C8—41no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—22no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95—41no—
167C24#12111yescharacteristic
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—41no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C4×D411yescharacteristic
197C22×C4C4×D411yescharacteristic
198C22×C4#12011yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4AGL(1,ℤ/8)11yescharacteristic
202C22×C4#13711yescharacteristic
203C22×C4#13711yescharacteristic
204C22×C4#13711yescharacteristic
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—81no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—82no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23#38611yescharacteristic
266C23#46811yescharacteristic
267C23#47011yescharacteristic
268Q8—42no—
269Q8—81no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—161no—
314C2×C4—161no—
315C2×C4—161no—
316C2×C4—161no—
317C2×C4#41711yescharacteristic
318C2×C4#41911yescharacteristic
319C2×C4#44311yescharacteristic
320C2×C4#44411yescharacteristic
321C2×C4#45811yescharacteristic
322C2×C4#45811yescharacteristic
323C2×C4#46811yescharacteristic
324C2×C4#47011yescharacteristic
325C8—42no—
326C8—81no—
327C8—81no—
328C8—81no—
329C8—81no—
330C8—81no—
331C8—161no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22#187611yescharacteristic
343C22#188111yescharacteristic
344C22#189011yescharacteristic
345C22#189411yescharacteristic
346C22#190511yescharacteristic
347C22#190611yescharacteristic
348C22#191011yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—162no—
364C4—161no—
365C4—161no—
366C2—41no—
367C2—41no—
368C2#903811yescharacteristic
369C2#920911yescharacteristic
370C2#928211yescharacteristic
371C2#930811yescharacteristic
372C2#937011yescharacteristic
373C2#947311yescharacteristic
374C2#952611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
10. #2100C4semidirect
11. #2099C4semidirect
12. #2096C4semidirect
13. #2095C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#276852
#223252
#87172
#87102
#85762
#70442
#70242